libstdc++
hypergeometric.tcc
1 // Special functions -*- C++ -*-
2 
3 // Copyright (C) 2006, 2007, 2008, 2009
4 // Free Software Foundation, Inc.
5 //
6 // This file is part of the GNU ISO C++ Library. This library is free
7 // software; you can redistribute it and/or modify it under the
8 // terms of the GNU General Public License as published by the
9 // Free Software Foundation; either version 3, or (at your option)
10 // any later version.
11 //
12 // This library is distributed in the hope that it will be useful,
13 // but WITHOUT ANY WARRANTY; without even the implied warranty of
14 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 // GNU General Public License for more details.
16 //
17 // Under Section 7 of GPL version 3, you are granted additional
18 // permissions described in the GCC Runtime Library Exception, version
19 // 3.1, as published by the Free Software Foundation.
20 
21 // You should have received a copy of the GNU General Public License and
22 // a copy of the GCC Runtime Library Exception along with this program;
23 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
24 // <http://www.gnu.org/licenses/>.
25 
26 /** @file tr1/hypergeometric.tcc
27  * This is an internal header file, included by other library headers.
28  * You should not attempt to use it directly.
29  */
30 
31 //
32 // ISO C++ 14882 TR1: 5.2 Special functions
33 //
34 
35 // Written by Edward Smith-Rowland based:
36 // (1) Handbook of Mathematical Functions,
37 // ed. Milton Abramowitz and Irene A. Stegun,
38 // Dover Publications,
39 // Section 6, pp. 555-566
40 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
41 
42 #ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
43 #define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1
44 
45 namespace std
46 {
47 namespace tr1
48 {
49 
50  // [5.2] Special functions
51 
52  // Implementation-space details.
53  namespace __detail
54  {
55 
56  /**
57  * @brief This routine returns the confluent hypergeometric function
58  * by series expansion.
59  *
60  * @f[
61  * _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)}
62  * \sum_{n=0}^{\infty}
63  * \frac{\Gamma(a+n)}{\Gamma(c+n)}
64  * \frac{x^n}{n!}
65  * @f]
66  *
67  * If a and b are integers and a < 0 and either b > 0 or b < a then the
68  * series is a polynomial with a finite number of terms. If b is an integer
69  * and b <= 0 the confluent hypergeometric function is undefined.
70  *
71  * @param __a The "numerator" parameter.
72  * @param __c The "denominator" parameter.
73  * @param __x The argument of the confluent hypergeometric function.
74  * @return The confluent hypergeometric function.
75  */
76  template<typename _Tp>
77  _Tp
78  __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x)
79  {
80  const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
81 
82  _Tp __term = _Tp(1);
83  _Tp __Fac = _Tp(1);
84  const unsigned int __max_iter = 100000;
85  unsigned int __i;
86  for (__i = 0; __i < __max_iter; ++__i)
87  {
88  __term *= (__a + _Tp(__i)) * __x
89  / ((__c + _Tp(__i)) * _Tp(1 + __i));
90  if (std::abs(__term) < __eps)
91  {
92  break;
93  }
94  __Fac += __term;
95  }
96  if (__i == __max_iter)
97  std::__throw_runtime_error(__N("Series failed to converge "
98  "in __conf_hyperg_series."));
99 
100  return __Fac;
101  }
102 
103 
104  /**
105  * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
106  * by an iterative procedure described in
107  * Luke, Algorithms for the Computation of Mathematical Functions.
108  *
109  * Like the case of the 2F1 rational approximations, these are
110  * probably guaranteed to converge for x < 0, barring gross
111  * numerical instability in the pre-asymptotic regime.
112  */
113  template<typename _Tp>
114  _Tp
115  __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin)
116  {
117  const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
118  const int __nmax = 20000;
119  const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
120  const _Tp __x = -__xin;
121  const _Tp __x3 = __x * __x * __x;
122  const _Tp __t0 = __a / __c;
123  const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c);
124  const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1)));
125  _Tp __F = _Tp(1);
126  _Tp __prec;
127 
128  _Tp __Bnm3 = _Tp(1);
129  _Tp __Bnm2 = _Tp(1) + __t1 * __x;
130  _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);
131 
132  _Tp __Anm3 = _Tp(1);
133  _Tp __Anm2 = __Bnm2 - __t0 * __x;
134  _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
135  + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;
136 
137  int __n = 3;
138  while(1)
139  {
140  _Tp __npam1 = _Tp(__n - 1) + __a;
141  _Tp __npcm1 = _Tp(__n - 1) + __c;
142  _Tp __npam2 = _Tp(__n - 2) + __a;
143  _Tp __npcm2 = _Tp(__n - 2) + __c;
144  _Tp __tnm1 = _Tp(2 * __n - 1);
145  _Tp __tnm3 = _Tp(2 * __n - 3);
146  _Tp __tnm5 = _Tp(2 * __n - 5);
147  _Tp __F1 = (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1);
148  _Tp __F2 = (_Tp(__n) + __a) * __npam1
149  / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
150  _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a)
151  / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
152  * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
153  _Tp __E = -__npam1 * (_Tp(__n - 1) - __c)
154  / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);
155 
156  _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
157  + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
158  _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
159  + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
160  _Tp __r = __An / __Bn;
161 
162  __prec = std::abs((__F - __r) / __F);
163  __F = __r;
164 
165  if (__prec < __eps || __n > __nmax)
166  break;
167 
168  if (std::abs(__An) > __big || std::abs(__Bn) > __big)
169  {
170  __An /= __big;
171  __Bn /= __big;
172  __Anm1 /= __big;
173  __Bnm1 /= __big;
174  __Anm2 /= __big;
175  __Bnm2 /= __big;
176  __Anm3 /= __big;
177  __Bnm3 /= __big;
178  }
179  else if (std::abs(__An) < _Tp(1) / __big
180  || std::abs(__Bn) < _Tp(1) / __big)
181  {
182  __An *= __big;
183  __Bn *= __big;
184  __Anm1 *= __big;
185  __Bnm1 *= __big;
186  __Anm2 *= __big;
187  __Bnm2 *= __big;
188  __Anm3 *= __big;
189  __Bnm3 *= __big;
190  }
191 
192  ++__n;
193  __Bnm3 = __Bnm2;
194  __Bnm2 = __Bnm1;
195  __Bnm1 = __Bn;
196  __Anm3 = __Anm2;
197  __Anm2 = __Anm1;
198  __Anm1 = __An;
199  }
200 
201  if (__n >= __nmax)
202  std::__throw_runtime_error(__N("Iteration failed to converge "
203  "in __conf_hyperg_luke."));
204 
205  return __F;
206  }
207 
208 
209  /**
210  * @brief Return the confluent hypogeometric function
211  * @f$ _1F_1(a;c;x) @f$.
212  *
213  * @todo Handle b == nonpositive integer blowup - return NaN.
214  *
215  * @param __a The "numerator" parameter.
216  * @param __c The "denominator" parameter.
217  * @param __x The argument of the confluent hypergeometric function.
218  * @return The confluent hypergeometric function.
219  */
220  template<typename _Tp>
221  inline _Tp
222  __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x)
223  {
224 #if _GLIBCXX_USE_C99_MATH_TR1
225  const _Tp __c_nint = std::tr1::nearbyint(__c);
226 #else
227  const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
228 #endif
229  if (__isnan(__a) || __isnan(__c) || __isnan(__x))
230  return std::numeric_limits<_Tp>::quiet_NaN();
231  else if (__c_nint == __c && __c_nint <= 0)
232  return std::numeric_limits<_Tp>::infinity();
233  else if (__a == _Tp(0))
234  return _Tp(1);
235  else if (__c == __a)
236  return std::exp(__x);
237  else if (__x < _Tp(0))
238  return __conf_hyperg_luke(__a, __c, __x);
239  else
240  return __conf_hyperg_series(__a, __c, __x);
241  }
242 
243 
244  /**
245  * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
246  * by series expansion.
247  *
248  * The hypogeometric function is defined by
249  * @f[
250  * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
251  * \sum_{n=0}^{\infty}
252  * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
253  * \frac{x^n}{n!}
254  * @f]
255  *
256  * This works and it's pretty fast.
257  *
258  * @param __a The first "numerator" parameter.
259  * @param __a The second "numerator" parameter.
260  * @param __c The "denominator" parameter.
261  * @param __x The argument of the confluent hypergeometric function.
262  * @return The confluent hypergeometric function.
263  */
264  template<typename _Tp>
265  _Tp
266  __hyperg_series(const _Tp __a, const _Tp __b,
267  const _Tp __c, const _Tp __x)
268  {
269  const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
270 
271  _Tp __term = _Tp(1);
272  _Tp __Fabc = _Tp(1);
273  const unsigned int __max_iter = 100000;
274  unsigned int __i;
275  for (__i = 0; __i < __max_iter; ++__i)
276  {
277  __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x
278  / ((__c + _Tp(__i)) * _Tp(1 + __i));
279  if (std::abs(__term) < __eps)
280  {
281  break;
282  }
283  __Fabc += __term;
284  }
285  if (__i == __max_iter)
286  std::__throw_runtime_error(__N("Series failed to converge "
287  "in __hyperg_series."));
288 
289  return __Fabc;
290  }
291 
292 
293  /**
294  * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
295  * by an iterative procedure described in
296  * Luke, Algorithms for the Computation of Mathematical Functions.
297  */
298  template<typename _Tp>
299  _Tp
300  __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c,
301  const _Tp __xin)
302  {
303  const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
304  const int __nmax = 20000;
305  const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
306  const _Tp __x = -__xin;
307  const _Tp __x3 = __x * __x * __x;
308  const _Tp __t0 = __a * __b / __c;
309  const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c);
310  const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2))
311  / (_Tp(2) * (__c + _Tp(1)));
312 
313  _Tp __F = _Tp(1);
314 
315  _Tp __Bnm3 = _Tp(1);
316  _Tp __Bnm2 = _Tp(1) + __t1 * __x;
317  _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);
318 
319  _Tp __Anm3 = _Tp(1);
320  _Tp __Anm2 = __Bnm2 - __t0 * __x;
321  _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
322  + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;
323 
324  int __n = 3;
325  while (1)
326  {
327  const _Tp __npam1 = _Tp(__n - 1) + __a;
328  const _Tp __npbm1 = _Tp(__n - 1) + __b;
329  const _Tp __npcm1 = _Tp(__n - 1) + __c;
330  const _Tp __npam2 = _Tp(__n - 2) + __a;
331  const _Tp __npbm2 = _Tp(__n - 2) + __b;
332  const _Tp __npcm2 = _Tp(__n - 2) + __c;
333  const _Tp __tnm1 = _Tp(2 * __n - 1);
334  const _Tp __tnm3 = _Tp(2 * __n - 3);
335  const _Tp __tnm5 = _Tp(2 * __n - 5);
336  const _Tp __n2 = __n * __n;
337  const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n
338  + _Tp(2) - __a * __b - _Tp(2) * (__a + __b))
339  / (_Tp(2) * __tnm3 * __npcm1);
340  const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n
341  + _Tp(2) - __a * __b) * __npam1 * __npbm1
342  / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
343  const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1
344  * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b))
345  / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
346  * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
347  const _Tp __E = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c)
348  / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);
349 
350  _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
351  + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
352  _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
353  + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
354  const _Tp __r = __An / __Bn;
355 
356  const _Tp __prec = std::abs((__F - __r) / __F);
357  __F = __r;
358 
359  if (__prec < __eps || __n > __nmax)
360  break;
361 
362  if (std::abs(__An) > __big || std::abs(__Bn) > __big)
363  {
364  __An /= __big;
365  __Bn /= __big;
366  __Anm1 /= __big;
367  __Bnm1 /= __big;
368  __Anm2 /= __big;
369  __Bnm2 /= __big;
370  __Anm3 /= __big;
371  __Bnm3 /= __big;
372  }
373  else if (std::abs(__An) < _Tp(1) / __big
374  || std::abs(__Bn) < _Tp(1) / __big)
375  {
376  __An *= __big;
377  __Bn *= __big;
378  __Anm1 *= __big;
379  __Bnm1 *= __big;
380  __Anm2 *= __big;
381  __Bnm2 *= __big;
382  __Anm3 *= __big;
383  __Bnm3 *= __big;
384  }
385 
386  ++__n;
387  __Bnm3 = __Bnm2;
388  __Bnm2 = __Bnm1;
389  __Bnm1 = __Bn;
390  __Anm3 = __Anm2;
391  __Anm2 = __Anm1;
392  __Anm1 = __An;
393  }
394 
395  if (__n >= __nmax)
396  std::__throw_runtime_error(__N("Iteration failed to converge "
397  "in __hyperg_luke."));
398 
399  return __F;
400  }
401 
402 
403  /**
404  * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection
405  * formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral
406  * and formula 15.3.11 for d = c - a - b integral.
407  * This assumes a, b, c != negative integer.
408  *
409  * The hypogeometric function is defined by
410  * @f[
411  * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
412  * \sum_{n=0}^{\infty}
413  * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
414  * \frac{x^n}{n!}
415  * @f]
416  *
417  * The reflection formula for nonintegral @f$ d = c - a - b @f$ is:
418  * @f[
419  * _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}
420  * _2F_1(a,b;1-d;1-x)
421  * + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}
422  * _2F_1(c-a,c-b;1+d;1-x)
423  * @f]
424  *
425  * The reflection formula for integral @f$ m = c - a - b @f$ is:
426  * @f[
427  * _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)}
428  * \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k}
429  * -
430  * @f]
431  */
432  template<typename _Tp>
433  _Tp
434  __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c,
435  const _Tp __x)
436  {
437  const _Tp __d = __c - __a - __b;
438  const int __intd = std::floor(__d + _Tp(0.5L));
439  const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
440  const _Tp __toler = _Tp(1000) * __eps;
441  const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max());
442  const bool __d_integer = (std::abs(__d - __intd) < __toler);
443 
444  if (__d_integer)
445  {
446  const _Tp __ln_omx = std::log(_Tp(1) - __x);
447  const _Tp __ad = std::abs(__d);
448  _Tp __F1, __F2;
449 
450  _Tp __d1, __d2;
451  if (__d >= _Tp(0))
452  {
453  __d1 = __d;
454  __d2 = _Tp(0);
455  }
456  else
457  {
458  __d1 = _Tp(0);
459  __d2 = __d;
460  }
461 
462  const _Tp __lng_c = __log_gamma(__c);
463 
464  // Evaluate F1.
465  if (__ad < __eps)
466  {
467  // d = c - a - b = 0.
468  __F1 = _Tp(0);
469  }
470  else
471  {
472 
473  bool __ok_d1 = true;
474  _Tp __lng_ad, __lng_ad1, __lng_bd1;
475  __try
476  {
477  __lng_ad = __log_gamma(__ad);
478  __lng_ad1 = __log_gamma(__a + __d1);
479  __lng_bd1 = __log_gamma(__b + __d1);
480  }
481  __catch(...)
482  {
483  __ok_d1 = false;
484  }
485 
486  if (__ok_d1)
487  {
488  /* Gamma functions in the denominator are ok.
489  * Proceed with evaluation.
490  */
491  _Tp __sum1 = _Tp(1);
492  _Tp __term = _Tp(1);
493  _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx
494  - __lng_ad1 - __lng_bd1;
495 
496  /* Do F1 sum.
497  */
498  for (int __i = 1; __i < __ad; ++__i)
499  {
500  const int __j = __i - 1;
501  __term *= (__a + __d2 + __j) * (__b + __d2 + __j)
502  / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x);
503  __sum1 += __term;
504  }
505 
506  if (__ln_pre1 > __log_max)
507  std::__throw_runtime_error(__N("Overflow of gamma functions "
508  "in __hyperg_luke."));
509  else
510  __F1 = std::exp(__ln_pre1) * __sum1;
511  }
512  else
513  {
514  // Gamma functions in the denominator were not ok.
515  // So the F1 term is zero.
516  __F1 = _Tp(0);
517  }
518  } // end F1 evaluation
519 
520  // Evaluate F2.
521  bool __ok_d2 = true;
522  _Tp __lng_ad2, __lng_bd2;
523  __try
524  {
525  __lng_ad2 = __log_gamma(__a + __d2);
526  __lng_bd2 = __log_gamma(__b + __d2);
527  }
528  __catch(...)
529  {
530  __ok_d2 = false;
531  }
532 
533  if (__ok_d2)
534  {
535  // Gamma functions in the denominator are ok.
536  // Proceed with evaluation.
537  const int __maxiter = 2000;
538  const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e();
539  const _Tp __psi_1pd = __psi(_Tp(1) + __ad);
540  const _Tp __psi_apd1 = __psi(__a + __d1);
541  const _Tp __psi_bpd1 = __psi(__b + __d1);
542 
543  _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1
544  - __psi_bpd1 - __ln_omx;
545  _Tp __fact = _Tp(1);
546  _Tp __sum2 = __psi_term;
547  _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx
548  - __lng_ad2 - __lng_bd2;
549 
550  // Do F2 sum.
551  int __j;
552  for (__j = 1; __j < __maxiter; ++__j)
553  {
554  // Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5
555  const _Tp __term1 = _Tp(1) / _Tp(__j)
556  + _Tp(1) / (__ad + __j);
557  const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1))
558  + _Tp(1) / (__b + __d1 + _Tp(__j - 1));
559  __psi_term += __term1 - __term2;
560  __fact *= (__a + __d1 + _Tp(__j - 1))
561  * (__b + __d1 + _Tp(__j - 1))
562  / ((__ad + __j) * __j) * (_Tp(1) - __x);
563  const _Tp __delta = __fact * __psi_term;
564  __sum2 += __delta;
565  if (std::abs(__delta) < __eps * std::abs(__sum2))
566  break;
567  }
568  if (__j == __maxiter)
569  std::__throw_runtime_error(__N("Sum F2 failed to converge "
570  "in __hyperg_reflect"));
571 
572  if (__sum2 == _Tp(0))
573  __F2 = _Tp(0);
574  else
575  __F2 = std::exp(__ln_pre2) * __sum2;
576  }
577  else
578  {
579  // Gamma functions in the denominator not ok.
580  // So the F2 term is zero.
581  __F2 = _Tp(0);
582  } // end F2 evaluation
583 
584  const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
585  const _Tp __F = __F1 + __sgn_2 * __F2;
586 
587  return __F;
588  }
589  else
590  {
591  // d = c - a - b not an integer.
592 
593  // These gamma functions appear in the denominator, so we
594  // catch their harmless domain errors and set the terms to zero.
595  bool __ok1 = true;
596  _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0);
597  _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0);
598  __try
599  {
600  __sgn_g1ca = __log_gamma_sign(__c - __a);
601  __ln_g1ca = __log_gamma(__c - __a);
602  __sgn_g1cb = __log_gamma_sign(__c - __b);
603  __ln_g1cb = __log_gamma(__c - __b);
604  }
605  __catch(...)
606  {
607  __ok1 = false;
608  }
609 
610  bool __ok2 = true;
611  _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0);
612  _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0);
613  __try
614  {
615  __sgn_g2a = __log_gamma_sign(__a);
616  __ln_g2a = __log_gamma(__a);
617  __sgn_g2b = __log_gamma_sign(__b);
618  __ln_g2b = __log_gamma(__b);
619  }
620  __catch(...)
621  {
622  __ok2 = false;
623  }
624 
625  const _Tp __sgn_gc = __log_gamma_sign(__c);
626  const _Tp __ln_gc = __log_gamma(__c);
627  const _Tp __sgn_gd = __log_gamma_sign(__d);
628  const _Tp __ln_gd = __log_gamma(__d);
629  const _Tp __sgn_gmd = __log_gamma_sign(-__d);
630  const _Tp __ln_gmd = __log_gamma(-__d);
631 
632  const _Tp __sgn1 = __sgn_gc * __sgn_gd * __sgn_g1ca * __sgn_g1cb;
633  const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a * __sgn_g2b;
634 
635  _Tp __pre1, __pre2;
636  if (__ok1 && __ok2)
637  {
638  _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
639  _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
640  + __d * std::log(_Tp(1) - __x);
641  if (__ln_pre1 < __log_max && __ln_pre2 < __log_max)
642  {
643  __pre1 = std::exp(__ln_pre1);
644  __pre2 = std::exp(__ln_pre2);
645  __pre1 *= __sgn1;
646  __pre2 *= __sgn2;
647  }
648  else
649  {
650  std::__throw_runtime_error(__N("Overflow of gamma functions "
651  "in __hyperg_reflect"));
652  }
653  }
654  else if (__ok1 && !__ok2)
655  {
656  _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
657  if (__ln_pre1 < __log_max)
658  {
659  __pre1 = std::exp(__ln_pre1);
660  __pre1 *= __sgn1;
661  __pre2 = _Tp(0);
662  }
663  else
664  {
665  std::__throw_runtime_error(__N("Overflow of gamma functions "
666  "in __hyperg_reflect"));
667  }
668  }
669  else if (!__ok1 && __ok2)
670  {
671  _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
672  + __d * std::log(_Tp(1) - __x);
673  if (__ln_pre2 < __log_max)
674  {
675  __pre1 = _Tp(0);
676  __pre2 = std::exp(__ln_pre2);
677  __pre2 *= __sgn2;
678  }
679  else
680  {
681  std::__throw_runtime_error(__N("Overflow of gamma functions "
682  "in __hyperg_reflect"));
683  }
684  }
685  else
686  {
687  __pre1 = _Tp(0);
688  __pre2 = _Tp(0);
689  std::__throw_runtime_error(__N("Underflow of gamma functions "
690  "in __hyperg_reflect"));
691  }
692 
693  const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d,
694  _Tp(1) - __x);
695  const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d,
696  _Tp(1) - __x);
697 
698  const _Tp __F = __pre1 * __F1 + __pre2 * __F2;
699 
700  return __F;
701  }
702  }
703 
704 
705  /**
706  * @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$.
707  *
708  * The hypogeometric function is defined by
709  * @f[
710  * _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
711  * \sum_{n=0}^{\infty}
712  * \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
713  * \frac{x^n}{n!}
714  * @f]
715  *
716  * @param __a The first "numerator" parameter.
717  * @param __a The second "numerator" parameter.
718  * @param __c The "denominator" parameter.
719  * @param __x The argument of the confluent hypergeometric function.
720  * @return The confluent hypergeometric function.
721  */
722  template<typename _Tp>
723  inline _Tp
724  __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
725  {
726 #if _GLIBCXX_USE_C99_MATH_TR1
727  const _Tp __a_nint = std::tr1::nearbyint(__a);
728  const _Tp __b_nint = std::tr1::nearbyint(__b);
729  const _Tp __c_nint = std::tr1::nearbyint(__c);
730 #else
731  const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L));
732  const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L));
733  const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
734 #endif
735  const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon();
736  if (std::abs(__x) >= _Tp(1))
737  std::__throw_domain_error(__N("Argument outside unit circle "
738  "in __hyperg."));
739  else if (__isnan(__a) || __isnan(__b)
740  || __isnan(__c) || __isnan(__x))
741  return std::numeric_limits<_Tp>::quiet_NaN();
742  else if (__c_nint == __c && __c_nint <= _Tp(0))
743  return std::numeric_limits<_Tp>::infinity();
744  else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler)
745  return std::pow(_Tp(1) - __x, __c - __a - __b);
746  else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0)
747  && __x >= _Tp(0) && __x < _Tp(0.995L))
748  return __hyperg_series(__a, __b, __c, __x);
749  else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10))
750  {
751  // For integer a and b the hypergeometric function is a finite polynomial.
752  if (__a < _Tp(0) && std::abs(__a - __a_nint) < __toler)
753  return __hyperg_series(__a_nint, __b, __c, __x);
754  else if (__b < _Tp(0) && std::abs(__b - __b_nint) < __toler)
755  return __hyperg_series(__a, __b_nint, __c, __x);
756  else if (__x < -_Tp(0.25L))
757  return __hyperg_luke(__a, __b, __c, __x);
758  else if (__x < _Tp(0.5L))
759  return __hyperg_series(__a, __b, __c, __x);
760  else
761  if (std::abs(__c) > _Tp(10))
762  return __hyperg_series(__a, __b, __c, __x);
763  else
764  return __hyperg_reflect(__a, __b, __c, __x);
765  }
766  else
767  return __hyperg_luke(__a, __b, __c, __x);
768  }
769 
770  } // namespace std::tr1::__detail
771 }
772 }
773 
774 #endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC